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Qin, Li; Huang, Haiyun; Ma, Yongge

Path integral and effective Hamiltonian in loop quantum cosmology. (English) Zbl 1266.83087

Gen. Relativ. Gravitation 45, No. 6, 1191-1210 (2013).
Summary: We study the path integral formulation of Friedmann universe filled with a massless scalar field in loop quantum cosmology. All the isotropic models of \(k=0,+1,-1\) are considered. To construct the path integrals in the timeless framework, a multiple group-averaging approach is proposed. Meanwhile, since the transition amplitude in the deparameterized framework can be expressed in terms of group-averaging, the path integrals can be formulated for both deparameterized and timeless frameworks. Their relation is clarified. It turns out that the effective Hamiltonian derived from the path integral in deparameterized framework is equivalent to the effective Hamiltonian constraint derived from the path integral in timeless framework, since they lead to same equations of motion. Moreover, the effective Hamiltonian constraints of above models derived in canonical theory are confirmed by the path integral formulation.

MSC:

83C45 Quantization of the gravitational field
83F05 Relativistic cosmology
83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
81S40 Path integrals in quantum mechanics

Keywords:

loop quantum cosmology; path integral; effective Hamiltonian; timeless framework; deparametrized framework
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